An Australian Queen Elizabeth would remain a common design

What would an Australian Queen Elizabeth-class battleship look like?

Essentially the same as the British ships, perhaps with improved air conditioning for warmer deployments.

Australia would receive essentially the same Queen Elizabeth design rather than a fundamentally different battleship. The value of the order would lie in joining a proven British class with shared weapons, machinery, training, and support. Better air conditioning for warmer seas is the one plausible local adaptation identified in the answer. Imperial commonality could matter more than creating a national variant.